Does one take the product rule or chain rule when there are 3 terms with variable being multiplied together.
example
taking
$$ x = r\cdot \cos \theta \cdot \sin \phi $$
at an instant in time
$$ x(t) = R(t) \cdot \cos\theta(t) \cdot \sin\phi(t) $$
to derive in order to obtain
$$ \dot{x}=R'(t) \cdot \cos'\theta(t) \cdot \sin'\phi(t) $$
I'm leaning towards product rule because
$$ f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) $$
but just doesn't seem to be coming out right.
I end up with
$$ \big(1 \cdot \cos\theta (t) \cdot \sin\phi (t)\big) + \big(R(t)\cdot (-\sin\theta (t) )\cdot \sin\phi (t)\big) + \big(R(t)\cos\theta (t) \cdot \cos\phi (t) \big) $$
Best Answer
If I am understanding correctly that $$x(t) = r(t)\cos(\theta(t))\sin(\phi(t))$$ then you will need both the product and chain rules to differentiate with respect to $t$. You will get \begin{align*} x'(t) &= r'(t)\cos(\theta(t))\sin(\phi(t)) - r(t)\sin(\theta(t))\theta'(t)\sin(\phi(t))\\[5pt] &\quad + r(t)\cos(\theta(t))\cos(\phi(t))\phi'(t). \end{align*}